Related papers: Variational approach to complicated similarity sol…
We discuss integrable extensions of real nonlinear wave equations with multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and…
We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non…
We analytically solve the one-dimensional coupled Gross-Pitaevskii equations which govern the motion of F=1 spinor Bose-Einstein condensates. The nonlinear density-density interactions are decoupled by making use of the unique properties of…
This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $\Omega$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that…
Via abstract results on maximal monotone operators and compactness property of Nemickii operator, existence of a weak solution for a class of nonlinear parabolic systems of partial differential equations is proven.
Selection of 25 examples from extensive nontrivial families for different types of nonlinear PDEs and their formal general solutions are given. The main goal here is to show on examples the types of solvable PDEs and what their general…
For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions $\cn(x,m)$ and…
Boundedness and blow-up of solutions for a nonlinear elliptic system arising in probability and stochastic processes
This article studies large solutions, for a class of quasi-linear equations involving p-Laplacian on the infinite cylindrical domains. We study the wellposedness of weak large solutions on infinite cylinders by the convergence of large…
Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by…
Fifth-order 1D nonlinear dispersion equations are shown to admit blow-up formation of shock waves as well as rarefaction waves. The concepts of smooth deformations are applied to distinguish "entropy" shocks from smooth rarefaction waves.…
We present n-dimensional vortex-ring-like and potential-like solutions with unusual properties related to some elliptical differential equations with compact sources. Solutions have almost 3- or 2-dimensional behaviour in the spaces with…
Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schr\"odinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their…
We present the exact bright one-soliton and two-soliton solutions of the integrable three coupled nonlinear Schroedinger equations (3-CNLS) by using the Hirota method, and then obtain them for the general $N$-coupled nonlinear Schroedinger…
In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we…
Leading order higher-spin corrections to the linearized higher-spin black brane are analyzed in four dimensions. It is shown that the static solution that respects planar symmetry exists in the bosonic case at given order. Its higher-spin…
We introduce a generalized similarity analysis which grants a qualitative description of the localised solutions of any nonlinear differential equation. This procedure provides relations between amplitude, width, and velocity of the…
The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink…
In a previous work, a perturbative approach to a class of Fokker-Planck equations, which have constant diffusion coefficients and small time-dependent drift coefficients, was developed by exploiting the close connection between the…
Differential inclusions with compact, upper semi-continuous, not necessarily convex right-hand sides in R^n are studied. Under a weakened monotonicity-type condition the existence of solutions is proved.